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[Thank you very much. I’m] Takanobu MIZUTA [from] SPARX Asset Management.
[I’m also belonging to] The University of Tokyo.
[Today, I’m going to talk about] Regulations' Effectiveness for Market Turbulence
by Large Erroneous Orders using Multi Agent Simulation
1
2
[This is] Today’s Talk, [table of contents].
[At first, I will describe] Motivation and our artificial market model (Multi-Agent
Simulation)
[In] Experiment 1, [we will investigate about] Characteristics of Erroneous-Order
Turbulence [and will show that]
Market prices declined not only during but also for a while after the period of
erroneous orders.
[And,] The amount of erroneous orders decided ranges of price falls.
[In] Experiment 2, [we will investigate about] Effects of Price Variation Limits to
the Turbulence [and will show that]
[The] Condition to Prevent the Large Turbulence [is], the limit time span should
be shorter than the time of erroneous orders existing.
[Lastly, I will mention] Summary & Implication to market regulations.
2
3
[First, I will describe] Motivation and our artificial market model.
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4
Motivation.
[In financial markets], large erroneous orders [sometimes] induce large price
fluctuations.
[Such] fluctuations [often cause] financial market turmoil.
For example, Flash Crush, in US, 2010.
[Therefore, there is] a big debate over which price regulations prevent [this
inducing].
[In] this study, [we build] Artificial Market Model (Multi-Agent Simulation) [to
investigate]
[Characteristics of Erroneous-Order Turbulence]
[And, Effects of] Price Regulations
This presentation only shows the case of “Price Variation Limit”
Although the proceeding shows other cases of price regulations.
4
5
Why Artificial Market Model?
Empirical Studies are very difficult to discuss such price regulations [because],
So many factors cause price formation in actual markets that an empirical study
cannot isolate the pure contribution of these regulations to price formation
[and because],
It is impossible to conduct experiments for new regulations in real financial
markets
[On the other hand], Artificial Market Model (Multi-Agent Simulation) can do
[them].
5
6
[This study's model is Similar to previous my presentation, previous technical
session.
I think, some people did not attend it. So, I will give similar talk here, again.]
[We built an artificial market model] on basis of Chiarella et. al. 2009.
●[Pricing mechanism is] Continuous Double Auction. [It is not simple, but, we
need] to implement realistic price variation limit
●Agent Model is Simple. [This is] to avoid arbitrary result, “Keep it short and
simple” [principle].
[We think Artificial Market Models should explain Stylized Facts as Simply as possible].
[There are] heterogeneous 1000 agents. [All agents calculate] Expected Return
[using this equation],
[And, the] strategy weights are different for each agent
・[First term is a] Fundamental [Strategy: When the market price is smaller than
the fundamental price, an agent expects a positive return , and vice verse].
・[Second term is a] technical [strategy: When historical return is positive, an
agent expects a positive return, and vice verse].
・[Third term is] noise.
[Plus, We implemented] Learning Process. [This is] different from Chiarella’s
study and previous my presentation
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7
[About] Learning Process,
Previous Our Study Mizuta et. al., 2013a Last year’s CIFEr (SSCI in Singapore)
[showed that]
Even thought without Learning Process
→ the model could replicate static characters of price variations.
→ however the model could NOT replicate Dynamic characters
of price variations, such as large fluctuations.
[therefore,] This Study also need Learning Process Because of treating large
price fluctuations
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8
Details of Learning Process
[Agents are] comparing Historical Return [and] each Strategy[’ term,
Fundamental strategy term, and Technical strategy term].
[When the strategy’s expected return and Historical Return are] Same Sign,
[This means] good performer Strategy.
[The strategy’s] Weight is Up.
[When the strategy’s expected return and Historical Return are] Opposite Sign,
[This means] bad performer Strategy.
[The strategy’s] Weight is Down.
[We also added] random learning.
In this way, agents learn better parameters and switch to the investment strategy
that estimates correctly.
9
[Next, I show simulation results of] Experiment 1 [about] Characteristics of
Erroneous-Order Turbulence
9
10
Model of Erroneous Orders
[There are] two parameters for the model: pg, density of erroneous orders, tg,
time of erroneous orders existing.
With a probability pg, His order is changed to Market Order Sell.
[Such market sell orders are] immediately done, and induce market price down.
This situation is maintained during tg.
11
[This figure shows] a time evolution of market prices with erroneous orders.
[We shade the time rage in which] erroneous orders exist.
Market prices declined not only during but also for a while after the period of
erroneous orders.
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12
[This figure shows] a time evolutions of strategies weights.
During erroneous-Order Existing switching Fundamental strategy to Technical
Strategy.
Because of increased technical strategy weight, market prices continued to
decline, even after the erroneous orders were gone,
12
13
[Next, we Measure] maximum Range of Price Declines and Reaching times to
the minimum price for various tg, pg under sg = tg times pg = const.
Maximum Range of Price Decline [is defined like this, from minimum market price
to initial market price].
Reaching times to the minimum price [is defined like this, from starting erroneous
orders to reaching time at minimum market price].
13
14
[This figure shows] maximum ranges of price declines for various tg, pg under sg
= tg times pg = const.
[Horizontal axe is] times of erroneous orders existing tg.
[The amount of erroneous orders, s_g on each line is constant.]
Ranges of price declines were almost the same when sg was constant, and
when, sg, was increasing more, the rages were increasing more.
[Therefore], The amount of erroneous orders, Sg, decided ranges of price falls,
[and] sg is a key parameter.
14
15
[This figure shows] reaching times to the minimum under sg = tg times pg =
const.
☆ Reaching times to the minimum price increase sufficiency less than
incrementation of tg.
☆ When sg was larger, reaching times were longer
15
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[Next, I show simulation results of] Experiment 2 [about] Effects of Price
Variation Limits to the Turbulence
16
17
[We modeled] the price variation limit [like this].
[There are] two constant parameters.
Tpl is a limit time span, and ⊿Ppl is limit price range.
[We referred] market price [Before tpl],
[and any agents] can Not order OutSide Pt-tpl ± ⊿Ppl
[I mean, Any] buy order prices above here, [they are] changed to this price.
[and any] sell order prices under here [they are] changed to this price.
18
[This figure shows a time evolution of market prices] with the price variation limit.
[In these parameters], The price variation limit succeeded to prevent the large
turbulences.
18
19
What is parameters’ Condition to Prevent Large Turbulence?
Large Erroneous Orders [have] 2 parameters: tg, pg
Price Variation Limits [have] 2 parameters:ΔPpl, tpl
We want to know
How should we set ΔPpl, tpl to prevent inducing
Large price fluctuation?
We measured Maximum Range of Price Declines
for various parameters, tg, pg, ΔPpl, tpl
19
20
Parameter Searching for 4 parameters is too heavy,
[Therefore, we] Reduce Searching Space Dimension 4 → 2
[As mentioned in] Experiment 1,
Characteristics of large erroneous orders are similar when Amount of erroneous
orders (Sg=tg×pg)= constant.
[Therefore, we] fix Sg.
[As mentioned in] Previous Our Study Mizuta et. al., 2013a,b
Price Variation Limits have the same effectiveness when Limit Price Range
(⊿Ppl) / Limit Time Span (tpl) = constant.
[Therefore, we] ⊿Ppl / tpl
Under these conditions,
We measured Maximum Range of Price Declines for various parameters, tg, pg,
ΔPpl, tpl
20
21
[This table lists] maximum ranges of price declines for various parameters.
Green Area [satisfy], tpl smaller than tg. [In these regions], Small Price Decline.
Condition to Prevent the Large Turbulence
[is the] limit time span should be shorter than [the] time of erroneous orders
existing.
21
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[Lastly, I will mention] Summary & Implication to market regulations.
22
23
[We built a simple] artificial market model to investigate effects of price variation
limits in large price fluctuation caused by large erroneous orders.
[In] Experiment 1, [we investigated about] Characteristics of Erroneous-Order
Turbulence [and showed that]
Market prices declined not only during but also for a while after the period of
erroneous orders.
[And], The amount of erroneous orders decided ranges of price falls.
[In] Experiment 2, [we investigated about] Effects of Price Variation Limits to the
Turbulence [and showed that]
[The] Condition to Prevent the Large Turbulence is, the limit time span should be
shorter than the time of erroneous orders existing.
23
24
[Lastly, I will mention] Implication [of this study result] to market regulations
In these days HFT (High Frequency Trading) are growing, very short time and
very dense amount erroneous-orders may happen.
[As I mentioned], price variation limit [should have] smaller limit time span [than]
time of erroneous orders existing.
This result implies that Price Variation Limit having very short tpl is needed.
[For Example], Tokyo Stock Exchange: two Price Variation Limits
○ “special quote”: tpl = 3 min
○ “daily price limits”: tpl = 1 bussiness day (5 hours)
[There limit time span may be] Too longer than time scale of HFT, [and they]
Need shorter tpl Price Variation Limit.
24
25
Could you say that again? (もう一度、おっしゃっていただけますか?)
I don’t quite understand your question. (ご質問の趣旨が良く分からないのですが)
Could you please rephrase your question? (ご質問を分かりやすく言い換えていただ
けますか)
So, you are asking me about.... (つまり、お尋ねの内容は...ですね)
I totally agree with you. (私も全くあなたと同意見です)
That’s a very challenging question for me to answer. (それは私にとって非常に答え
がいのある質問です)
That’s a question I’m not sure I can answer right now. (そのご質問にすぐお答えで
きるかどうか分かりません)
It would require further research. (さらなる研究結果を待ちたい)
You are right on that point. (その点に関してはあなたが正しい)
Our method will not solve the problem. (我々の方法ではその問題は解決できない)
25
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2727
28
We built an artificial market model on basis of Chiarella et. al. [2009].
Chiarella’s model did not include Learning Process, however,
We built Learning Process of agents.
And we are comparing between the case with Learning Process and without it.
Our Agent Model is Simple. This is to avoid arbitrary result, “Keep it short and
simple” principle.
We think Artificial Market Models should explain Stylized Facts as Simply as
possible,
Our pricing mechanism is Continuous Double Auction
It is not simple, but, we need to implement realistic price variation limit
Learning process
Here, Learning process means agents are switching strategy, fundamental
strategy or technical strategy.
We will show that, an Overshoot occurred in the case With the learning process,
however, overshoot did not occur in the case WithOut the process
28
29
Next, I will describe agent model.
All agents calculate Expected Return using this equation.
First term is a Fundamental Strategy:
When the market price is smaller than the fundamental price, an agent expects a
positive return , and vice verse.
Second term is a technical strategy:
When historical return is positive, an agent expects a positive return, and vice
verse.
Third term is noise,
After the expected return has been determined, an expected price is determined
like this.
And, agents order base on this Expected Price.
30
Next, agents determine order price and, buy or sell.
To Stabilize simulation runs for the continuous double mechanism,
Order Prices must be covered widely in Order Book.
We modeled an Order Price, Po, by Random variables of Uniformly distributed in
the interval from Expected Price, Pe, minus constant, Pd, to Pe plus Pd.
And then,
When Po lager than Pe, the agent orders to sell one unit.
When Po smaller than Pe, the agent orders to buy one unit.
31
This Table lists Traditional stylized facts in each case.
In all cases, both kurtosis and autocorrelation for square returns for all i are
positive.
This means that all cases replicate Traditional stylized facts: fat-tail and volatility-
clustering.
31
3232
3333
34
We examined two Cases, Case 1, Fundamental Value is constant,
Case 2, Fundamental value is rapid incremented like this. This is bubble inducing
trigger.
For Each cases, we examined With learning process And WithOut learning
process.
Therefore, we examined four cases in all.
35
This Figure shows time evolution of market prices in case 1, Fundamental Value
is constant.
In both cases, With learning process and without learning process. the results are
very similar,
The prices were small fluctuating around Fundamental Value, here, Ten
Thousand
35
36
This Figure shows time evolution of prices in case 2.
Fundamental value was changed at this time, increased to New Fundamental
Value, Fifteen Thousand.
This is the bubble inducing trigger.
Without Learning Process, Black line, Overshooting was not occurred.
On the other hand, with Learning Process, Red line,
the price went far beyond the new fundamental value.
Only with learning process, Overshoot occurred.
36
37
This Table lists Traditional stylized facts in each case.
In all cases, both kurtosis and autocorrelation for square returns for all i are
positive.
This means that all cases replicate Traditional stylized facts: fat-tail and volatility-
clustering.
37
38
We propose Hazard Rate as New Stylized fact to verify model replicating
overshoot
Hazard Rate Hi is conditional probability that sequence of positive return ends at
i, given that it lasts until i.
For Example i=3, H3 means like this.
1st, positive return, 2nd, positive, 3rd positive,
In this condition, H3 is probability of 4th return become negative.
Empirical Studies showed that, Any cases, Hi for most of i are smaller than 50%
And when including overshoot period, Hi decline rapidly with i,
This show that the overshoot returns tend to continue to be positive
And this tendency stronger continuing positive returns longer
38
39
This Table lists New Stylized Facts: Hazard Rate in each case.
In case 2 with learning, hazard rate declined rapidly.
This case can replicate a significant Overshoot like actual markets.
On the other hand, the case without learning, hazard rate dose not declined
rapidly.
The case can not replicate Overshoot.
Case 1, without learning, Hazard rates are upper 50% for all i.
This is Not consistent with empirical study.
On the other hand, Case 1, with learning, Hazard rates for most of i are smaller
than 50%, even when price fluctuations are stable.
This consistent with empirical study.
Therefore, only cases with Learning Process were verified by Hazard Rate,
and only Case 2 can replicate overshoot.
39
40
Result Summary Experiment 1 relationship between Learning process and
replicating Overshoot
The cases With learning process, both case 1 and case 2, were Consistent with
Empirical study verified by Hazard Rate.
And case 2 can replicate overshoot, bubble and crush
The cases Without Learning Process were Not consistent with Empirical study
Not verified by Hazard Rate.
40

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Regulations' Effectiveness for Market Turbulence by Large Erroneous Orders using Multi Agent Simulation

  • 1. 1 [Thank you very much. I’m] Takanobu MIZUTA [from] SPARX Asset Management. [I’m also belonging to] The University of Tokyo. [Today, I’m going to talk about] Regulations' Effectiveness for Market Turbulence by Large Erroneous Orders using Multi Agent Simulation 1
  • 2. 2 [This is] Today’s Talk, [table of contents]. [At first, I will describe] Motivation and our artificial market model (Multi-Agent Simulation) [In] Experiment 1, [we will investigate about] Characteristics of Erroneous-Order Turbulence [and will show that] Market prices declined not only during but also for a while after the period of erroneous orders. [And,] The amount of erroneous orders decided ranges of price falls. [In] Experiment 2, [we will investigate about] Effects of Price Variation Limits to the Turbulence [and will show that] [The] Condition to Prevent the Large Turbulence [is], the limit time span should be shorter than the time of erroneous orders existing. [Lastly, I will mention] Summary & Implication to market regulations. 2
  • 3. 3 [First, I will describe] Motivation and our artificial market model. 3
  • 4. 4 Motivation. [In financial markets], large erroneous orders [sometimes] induce large price fluctuations. [Such] fluctuations [often cause] financial market turmoil. For example, Flash Crush, in US, 2010. [Therefore, there is] a big debate over which price regulations prevent [this inducing]. [In] this study, [we build] Artificial Market Model (Multi-Agent Simulation) [to investigate] [Characteristics of Erroneous-Order Turbulence] [And, Effects of] Price Regulations This presentation only shows the case of “Price Variation Limit” Although the proceeding shows other cases of price regulations. 4
  • 5. 5 Why Artificial Market Model? Empirical Studies are very difficult to discuss such price regulations [because], So many factors cause price formation in actual markets that an empirical study cannot isolate the pure contribution of these regulations to price formation [and because], It is impossible to conduct experiments for new regulations in real financial markets [On the other hand], Artificial Market Model (Multi-Agent Simulation) can do [them]. 5
  • 6. 6 [This study's model is Similar to previous my presentation, previous technical session. I think, some people did not attend it. So, I will give similar talk here, again.] [We built an artificial market model] on basis of Chiarella et. al. 2009. ●[Pricing mechanism is] Continuous Double Auction. [It is not simple, but, we need] to implement realistic price variation limit ●Agent Model is Simple. [This is] to avoid arbitrary result, “Keep it short and simple” [principle]. [We think Artificial Market Models should explain Stylized Facts as Simply as possible]. [There are] heterogeneous 1000 agents. [All agents calculate] Expected Return [using this equation], [And, the] strategy weights are different for each agent ・[First term is a] Fundamental [Strategy: When the market price is smaller than the fundamental price, an agent expects a positive return , and vice verse]. ・[Second term is a] technical [strategy: When historical return is positive, an agent expects a positive return, and vice verse]. ・[Third term is] noise. [Plus, We implemented] Learning Process. [This is] different from Chiarella’s study and previous my presentation 6
  • 7. 7 [About] Learning Process, Previous Our Study Mizuta et. al., 2013a Last year’s CIFEr (SSCI in Singapore) [showed that] Even thought without Learning Process → the model could replicate static characters of price variations. → however the model could NOT replicate Dynamic characters of price variations, such as large fluctuations. [therefore,] This Study also need Learning Process Because of treating large price fluctuations 7
  • 8. 8 Details of Learning Process [Agents are] comparing Historical Return [and] each Strategy[’ term, Fundamental strategy term, and Technical strategy term]. [When the strategy’s expected return and Historical Return are] Same Sign, [This means] good performer Strategy. [The strategy’s] Weight is Up. [When the strategy’s expected return and Historical Return are] Opposite Sign, [This means] bad performer Strategy. [The strategy’s] Weight is Down. [We also added] random learning. In this way, agents learn better parameters and switch to the investment strategy that estimates correctly.
  • 9. 9 [Next, I show simulation results of] Experiment 1 [about] Characteristics of Erroneous-Order Turbulence 9
  • 10. 10 Model of Erroneous Orders [There are] two parameters for the model: pg, density of erroneous orders, tg, time of erroneous orders existing. With a probability pg, His order is changed to Market Order Sell. [Such market sell orders are] immediately done, and induce market price down. This situation is maintained during tg.
  • 11. 11 [This figure shows] a time evolution of market prices with erroneous orders. [We shade the time rage in which] erroneous orders exist. Market prices declined not only during but also for a while after the period of erroneous orders. 11
  • 12. 12 [This figure shows] a time evolutions of strategies weights. During erroneous-Order Existing switching Fundamental strategy to Technical Strategy. Because of increased technical strategy weight, market prices continued to decline, even after the erroneous orders were gone, 12
  • 13. 13 [Next, we Measure] maximum Range of Price Declines and Reaching times to the minimum price for various tg, pg under sg = tg times pg = const. Maximum Range of Price Decline [is defined like this, from minimum market price to initial market price]. Reaching times to the minimum price [is defined like this, from starting erroneous orders to reaching time at minimum market price]. 13
  • 14. 14 [This figure shows] maximum ranges of price declines for various tg, pg under sg = tg times pg = const. [Horizontal axe is] times of erroneous orders existing tg. [The amount of erroneous orders, s_g on each line is constant.] Ranges of price declines were almost the same when sg was constant, and when, sg, was increasing more, the rages were increasing more. [Therefore], The amount of erroneous orders, Sg, decided ranges of price falls, [and] sg is a key parameter. 14
  • 15. 15 [This figure shows] reaching times to the minimum under sg = tg times pg = const. ☆ Reaching times to the minimum price increase sufficiency less than incrementation of tg. ☆ When sg was larger, reaching times were longer 15
  • 16. 16 [Next, I show simulation results of] Experiment 2 [about] Effects of Price Variation Limits to the Turbulence 16
  • 17. 17 [We modeled] the price variation limit [like this]. [There are] two constant parameters. Tpl is a limit time span, and ⊿Ppl is limit price range. [We referred] market price [Before tpl], [and any agents] can Not order OutSide Pt-tpl ± ⊿Ppl [I mean, Any] buy order prices above here, [they are] changed to this price. [and any] sell order prices under here [they are] changed to this price.
  • 18. 18 [This figure shows a time evolution of market prices] with the price variation limit. [In these parameters], The price variation limit succeeded to prevent the large turbulences. 18
  • 19. 19 What is parameters’ Condition to Prevent Large Turbulence? Large Erroneous Orders [have] 2 parameters: tg, pg Price Variation Limits [have] 2 parameters:ΔPpl, tpl We want to know How should we set ΔPpl, tpl to prevent inducing Large price fluctuation? We measured Maximum Range of Price Declines for various parameters, tg, pg, ΔPpl, tpl 19
  • 20. 20 Parameter Searching for 4 parameters is too heavy, [Therefore, we] Reduce Searching Space Dimension 4 → 2 [As mentioned in] Experiment 1, Characteristics of large erroneous orders are similar when Amount of erroneous orders (Sg=tg×pg)= constant. [Therefore, we] fix Sg. [As mentioned in] Previous Our Study Mizuta et. al., 2013a,b Price Variation Limits have the same effectiveness when Limit Price Range (⊿Ppl) / Limit Time Span (tpl) = constant. [Therefore, we] ⊿Ppl / tpl Under these conditions, We measured Maximum Range of Price Declines for various parameters, tg, pg, ΔPpl, tpl 20
  • 21. 21 [This table lists] maximum ranges of price declines for various parameters. Green Area [satisfy], tpl smaller than tg. [In these regions], Small Price Decline. Condition to Prevent the Large Turbulence [is the] limit time span should be shorter than [the] time of erroneous orders existing. 21
  • 22. 22 [Lastly, I will mention] Summary & Implication to market regulations. 22
  • 23. 23 [We built a simple] artificial market model to investigate effects of price variation limits in large price fluctuation caused by large erroneous orders. [In] Experiment 1, [we investigated about] Characteristics of Erroneous-Order Turbulence [and showed that] Market prices declined not only during but also for a while after the period of erroneous orders. [And], The amount of erroneous orders decided ranges of price falls. [In] Experiment 2, [we investigated about] Effects of Price Variation Limits to the Turbulence [and showed that] [The] Condition to Prevent the Large Turbulence is, the limit time span should be shorter than the time of erroneous orders existing. 23
  • 24. 24 [Lastly, I will mention] Implication [of this study result] to market regulations In these days HFT (High Frequency Trading) are growing, very short time and very dense amount erroneous-orders may happen. [As I mentioned], price variation limit [should have] smaller limit time span [than] time of erroneous orders existing. This result implies that Price Variation Limit having very short tpl is needed. [For Example], Tokyo Stock Exchange: two Price Variation Limits ○ “special quote”: tpl = 3 min ○ “daily price limits”: tpl = 1 bussiness day (5 hours) [There limit time span may be] Too longer than time scale of HFT, [and they] Need shorter tpl Price Variation Limit. 24
  • 25. 25 Could you say that again? (もう一度、おっしゃっていただけますか?) I don’t quite understand your question. (ご質問の趣旨が良く分からないのですが) Could you please rephrase your question? (ご質問を分かりやすく言い換えていただ けますか) So, you are asking me about.... (つまり、お尋ねの内容は...ですね) I totally agree with you. (私も全くあなたと同意見です) That’s a very challenging question for me to answer. (それは私にとって非常に答え がいのある質問です) That’s a question I’m not sure I can answer right now. (そのご質問にすぐお答えで きるかどうか分かりません) It would require further research. (さらなる研究結果を待ちたい) You are right on that point. (その点に関してはあなたが正しい) Our method will not solve the problem. (我々の方法ではその問題は解決できない) 25
  • 26. 2626
  • 27. 2727
  • 28. 28 We built an artificial market model on basis of Chiarella et. al. [2009]. Chiarella’s model did not include Learning Process, however, We built Learning Process of agents. And we are comparing between the case with Learning Process and without it. Our Agent Model is Simple. This is to avoid arbitrary result, “Keep it short and simple” principle. We think Artificial Market Models should explain Stylized Facts as Simply as possible, Our pricing mechanism is Continuous Double Auction It is not simple, but, we need to implement realistic price variation limit Learning process Here, Learning process means agents are switching strategy, fundamental strategy or technical strategy. We will show that, an Overshoot occurred in the case With the learning process, however, overshoot did not occur in the case WithOut the process 28
  • 29. 29 Next, I will describe agent model. All agents calculate Expected Return using this equation. First term is a Fundamental Strategy: When the market price is smaller than the fundamental price, an agent expects a positive return , and vice verse. Second term is a technical strategy: When historical return is positive, an agent expects a positive return, and vice verse. Third term is noise, After the expected return has been determined, an expected price is determined like this. And, agents order base on this Expected Price.
  • 30. 30 Next, agents determine order price and, buy or sell. To Stabilize simulation runs for the continuous double mechanism, Order Prices must be covered widely in Order Book. We modeled an Order Price, Po, by Random variables of Uniformly distributed in the interval from Expected Price, Pe, minus constant, Pd, to Pe plus Pd. And then, When Po lager than Pe, the agent orders to sell one unit. When Po smaller than Pe, the agent orders to buy one unit.
  • 31. 31 This Table lists Traditional stylized facts in each case. In all cases, both kurtosis and autocorrelation for square returns for all i are positive. This means that all cases replicate Traditional stylized facts: fat-tail and volatility- clustering. 31
  • 32. 3232
  • 33. 3333
  • 34. 34 We examined two Cases, Case 1, Fundamental Value is constant, Case 2, Fundamental value is rapid incremented like this. This is bubble inducing trigger. For Each cases, we examined With learning process And WithOut learning process. Therefore, we examined four cases in all.
  • 35. 35 This Figure shows time evolution of market prices in case 1, Fundamental Value is constant. In both cases, With learning process and without learning process. the results are very similar, The prices were small fluctuating around Fundamental Value, here, Ten Thousand 35
  • 36. 36 This Figure shows time evolution of prices in case 2. Fundamental value was changed at this time, increased to New Fundamental Value, Fifteen Thousand. This is the bubble inducing trigger. Without Learning Process, Black line, Overshooting was not occurred. On the other hand, with Learning Process, Red line, the price went far beyond the new fundamental value. Only with learning process, Overshoot occurred. 36
  • 37. 37 This Table lists Traditional stylized facts in each case. In all cases, both kurtosis and autocorrelation for square returns for all i are positive. This means that all cases replicate Traditional stylized facts: fat-tail and volatility- clustering. 37
  • 38. 38 We propose Hazard Rate as New Stylized fact to verify model replicating overshoot Hazard Rate Hi is conditional probability that sequence of positive return ends at i, given that it lasts until i. For Example i=3, H3 means like this. 1st, positive return, 2nd, positive, 3rd positive, In this condition, H3 is probability of 4th return become negative. Empirical Studies showed that, Any cases, Hi for most of i are smaller than 50% And when including overshoot period, Hi decline rapidly with i, This show that the overshoot returns tend to continue to be positive And this tendency stronger continuing positive returns longer 38
  • 39. 39 This Table lists New Stylized Facts: Hazard Rate in each case. In case 2 with learning, hazard rate declined rapidly. This case can replicate a significant Overshoot like actual markets. On the other hand, the case without learning, hazard rate dose not declined rapidly. The case can not replicate Overshoot. Case 1, without learning, Hazard rates are upper 50% for all i. This is Not consistent with empirical study. On the other hand, Case 1, with learning, Hazard rates for most of i are smaller than 50%, even when price fluctuations are stable. This consistent with empirical study. Therefore, only cases with Learning Process were verified by Hazard Rate, and only Case 2 can replicate overshoot. 39
  • 40. 40 Result Summary Experiment 1 relationship between Learning process and replicating Overshoot The cases With learning process, both case 1 and case 2, were Consistent with Empirical study verified by Hazard Rate. And case 2 can replicate overshoot, bubble and crush The cases Without Learning Process were Not consistent with Empirical study Not verified by Hazard Rate. 40